Metrics on Bundle Spaces and Harmonic
Gauss MapsDoctoral Dissertation, December 1991 |

The aim of this work is to define a pseudoriemannian metric on the
Grassmannian bundles Gover a pseudoriemannian
manifold _{n }M M, with n < dim M+1, and to
explore the relationship between the harmonicity of an immersion in
M and the harmonicity of its Gauss map into
G. In the process, a general method is developed for
defining, à la Sasaki, a metric on the total space of a fiber
bundle, given metrics on the base space and on the standard fiber, and a
connection on the bundle. This metric exists only when the structure
group of the bundle is reducible to a subgroup of the isotropy group of
the metric on the standard fiber. In this case, the metric obtained also
depends on how this reduction of the structure group is achieved. _{n }M
Once the metric is defined, two pseudoriemannian connections on the total
space are considered. One is the Levi-Civita connection. The other is
also pseudoriemannian, and is closely related to the connections on the
base space and the standard fiber. The relationship between the
harmonicity of a map |

A Historical Overview of Connections in
Geometry Master's Thesis, May 2011 |

This thesis is an attempt to untangle/clarify the modern theory of
connections in Geometry. Towards this end a historical approach was taken
and original as well as secondary sources were used. An overview of the
most important historical developments is given as well as a modern look
at how the various definitions of connection are related.
I hope to clear up some of the confusions surrounding a connection in
Differential Geometry; or, at least some of the things that confused me
when I was trying to figure out just what exactly a connection is. In
particular, 1) How is a covariant derivative operator related to a
connection? 2)How is parallel transport related to the previous two
notions? 3) What was the first definition of a connection? 4) What is a
connection in the most general sense? I hope that I answer all of these
questions satisfactorily in the following pages. I have also provided a
chart of the heirarchy of connections |

Geometry of Horizontal Bundles and
ConnectionsDoctoral Dissertation, May 2014 |

An Ehresmann connection on a fiber bundle E over M is defined
by prescribing a suitable horizontal subbundle H of the tangent
bundle TE over E. For a horizontal bundle to be suitable, it
must have the property of horizontal path lifting. This ensures that the
horizontal bundle determines a system of parallel transport between any two
fibers of E.
The main result of this dissertation is a geometric characterization of
the horizontal bundles on In order for a horizontal bundle to admit a system of parallel transport
or have holonomy, it must be a connection. However, certain other geometric
properties that are usually attributed to connections are actually
properties of arbitrary horizontal bundles. These properties are studied in
the case when |